Booleans
type
logical

The two main boolean values are true and false. They stand for the logical values of the same name. They appear as values of the conditions in if-statements and while-loops. Booleans are also important as return values of filters (see ) such as and . Note that it is a convention that the name of a function that returns true or false according to the outcome, starts with Is.

For technical reasons, also the value fail (see ) is regarded as a boolean.

IsBool (Filter)
<#Include Label="IsBool">

Fail (Variable)
The value fail is used to indicate situations when an operation could not be performed for the given arguments, either because of shortcomings of the arguments or because of restrictions in the implementation or computability. So for example will return fail if the point searched for is not in the list.

fail
is simply an object that is different from every other object than itself.

For technical reasons, fail is a boolean value. But note that fail cannot be used to form boolean expressions with and, or, and not (see below), and fail cannot appear in boolean lists (see Chapter ).
Comparisons of Booleans
comparisons
Equality and inequality of Booleans
equality
inequality
bool1 = bool2

\noindent
bool1 <> bool2

The equality operator = evaluates to true if the two boolean values bool1 and bool2 are equal, i.e., both are true or both are false or both fail, and false otherwise. The inequality operator <> evaluates to true if the two boolean values bool1, bool2 are different, and false otherwise. This operation is also called the exclusive or, because its value is true if exactly one of bool1 or bool2 is true.

You can compare boolean values with objects of other types. Of course they are never equal.

gap> true = false; false gap> false = (true = fail); true gap> true <> 17; true
Ordering of Booleans
ordering
bool1 < bool2

The ordering of boolean values is defined by true < false < fail. For the comparison of booleans with other &GAP; objects, see Section .

gap> true < false; fail >= false; true true
Operations for Booleans
operations
logical operations

The following boolean operations are only applicable to true and false.

Logical disjunction
Logical disjunction
or
bool1 or bool2

The logical operator or evaluates to true if at least one of the two boolean operands bool1 and bool2 is true, and to false otherwise.

or
first evaluates bool1. If the value is neither true nor false an error is signalled. If the value is true, then or returns true without evaluating bool2. If the value is false, then or evaluates bool2. Again, if the value is neither true nor false an error is signalled. Otherwise or returns the value of bool2. This short-circuited evaluation is important if the value of bool1 is true and evaluation of bool2 would take much time or cause an error.

or
is associative, i.e., it is allowed to write b1 or b2 or b3, which is interpreted as (b1 or b2) or b3. or has the lowest precedence of the logical operators. All logical operators have lower precedence than the comparison operators =, <, in, etc.

gap> true or false; true gap> false or false; false gap> i := -1;; l := [1,2,3];; gap> if i <= 0 or l[i] = false then # this does not cause an error, > Print("aha\n"); fi; # because `l[i]' is not evaluated aha
Logical conjunction
Logical conjunction
and
bool1 and bool2

and
\noindent
fil1 and fil2

The logical operator and evaluates to true if both boolean operands bool1, bool2 are true, and to false otherwise.

and
first evaluates bool1. If the value is neither true nor false an error is signalled. If the value is false, then and returns false without evaluating bool2. If the value is true, then and evaluates bool2. Again, if the value is neither true nor false an error is signalled. Otherwise and returns the value of bool2. This short-circuited evaluation is important if the value of bool1 is false and evaluation of bool2 would take much time or cause an error.

and
is associative, i.e., it is allowed to write b1 and b2 and b3, which is interpreted as (b1 and b2) and b3. and has higher precedence than the logical or operator, but lower than the unary logical not operator. All logical operators have lower precedence than the comparison operators =, <, in, etc.

gap> true and false; false gap> true and true; true gap> false and 17; # does not cause error, because 17 is never looked at false

and
can also be applied to filters. It returns a filter that when applied to some argument x, tests fil1(x) and fil2(x).

gap> andfilt:= IsPosRat and IsInt;; gap> andfilt( 17 ); andfilt( 1/2 ); true false
Logical negation
Logical negation
not
not
bool

The logical operator not returns true if the boolean value bool is false, and true otherwise. An error is signalled if bool does not evaluate to true or false.

not
has higher precedence than the other logical operators, or and and. All logical operators have lower precedence than the comparison operators =, <, in, etc.

gap> true and false; false gap> not true; false gap> not false; true